Give me a child until the age of seven, the Jesuits used to argue, and I will have him for life. Primary education has always been an ideological battlefield, because it's what happens in those formative years that determines not just the path of further education, but the course of a child's life.

Lady Plowden's report in the 1960s on the future of the primary curriculum still divides opinion. It was greeted warmly by the educational establishment of its time, but there were also voices casting doubt on the approach it took, primarily from those who feared that the vital importance of knowledge was being downgraded. The most vehement voices against the Plowden approach were on the right, but concern about Plowden's approach was most powerfully articulated by Jim Callaghan, the Labour prime minister. In his famous Ruskin speech, he sought to start a "great debate" on education to enlist public opinion on his side against what he saw as the capture of educational orthodoxy by an unrepresentative elite.

Now, four decades after Plowden, comes Professor Robin Alexander. Another academic exercise in analysing primary education has sharply divided opinion. It's well worth everyone engaged with the education debate spending time reading this Cambridge University-led review. Its analysis is thought-provoking and provocative. But its recommendations do not always convince.

Overall, the picture of primary education that it paints reflects the resilience and achievements of a generation of primary school teachers who have worked harder than ever to raise attainment and standards. The Cambridge review is rightly sympathetic to teachers who feel their professional status is undermined by a bureaucratic burden that seems to increase with every week that passes.

I sympathise with them, too. The amount of guidance sent out by the Department for Children, Schools and Families each year amounts to 4,000 pages of documentation – longer than the King James Bible.

At the heart of improving education is the quality of teaching. Countries with the best teachers routinely come top of the international league tables. Our current generation of teachers is superb. But we must always strive to improve.

Academic research shows that the quality of teachers is the most important factor in a child's education, which is why we must encourage the brightest graduates into the classroom. I want to give teachers the chance to deepen their knowledge of their subjects by making it easier for them to pursue higher-level qualifications.

One particularly strong argument of the review is the emphasis on the importance of subject specialists at key stage 2. In that respect, the report can be seen as a rebuke to the idea put forward by the Rose review this year that subjects be collapsed into cross-cutting themes. Some of the best primary schools in the country, such as Durand in Lambeth, London, insist on discrete subject teaching in KS2 and also stream children, giving them a broad curriculum, personalised education and high standards. That half of the children at Durand are eligible for free school meals and a third of its 11-year-olds reach level 5 at key stage 2 shows that a broad and demanding curriculum – far from undermining reading, writing and arithmetic – reinforces attainment in these core skills.

However, while the latest review may have something to tell us about the curriculum and about professional autonomy overall, two areas have caught the public eye where I fear he is wrong. They are proposals to raise the age of compulsory schooling and make accountability fuzzier. One of our central concerns about primary education is the way in which poorer children fall behind by the time they leave at 11. The lesson from the brilliant Knowledge is Power Programme schools in America is that the most disadvantaged children benefit most from more schooling, not less.

The Cambridge review's critique of the current system of testing clearly struck a chord with some. But we need to maintain accountability. As well as being able to recognise which schools are weaker, so we can see where more help is needed, we need to identify those at the top so they can be emulated. The schools I've been to that are most enthusiastic about league tables are not enclaves of privilege but outstanding schools in poorer areas that were once written off because of their intake. Often these schools are outperforming others in their area. It's by learning from those sorts of schools, and Durand is one, that we can ensure the spread of best practice.

Ultimately, we need to give parents a bigger role in shaping their children's education, because the greater the parental involvement, the higher the level of attainment. For parents to be meaningfully involved, they need information about how schools are doing – and that accountability needs to be sharper.

The arguments over primary education won't end with the publication of this review. But if we can build a consensus that we need less bureaucracy, while respecting the need for proper accountability, and a truly stretching curriculum, I'm convinced we can progress towards a primary education system that provides the best possible opportunities for every child.